Blender Game Engine not displaying World texture
I can't actually find anything which says that sky textures are unsupported in the game engine, but there are a lot if things which are unsupported in the game engine..

One thing you could do instead is use a skydome, which is basically a UV sphere  with a sky texture mapped to it.


Add a UV sphere and scale it up so it covers your entire scene.
In edit mode, select everything (A) and press &#9096; Ctrl&#8679; ShiftN to flip the normals.
To avoid extreme distortion around the poles, you can make infinitely small holes in the top and bottom of the sphere.
In top view (Numpad 7), select the polar vertices. Press &#9096; Ctrl&#8679; ShiftB and type in a small value like .001. Press &#9166; Enter and &#8998; Delete>Faces to delete the new faces at the poles (creating holes), then press S 0 to scale the holes to infinitely small points. 

Make sure the vertices are not merged when you do this, otherwise you'll be back where you started.
From side view (Numpad 1), press U> Sphere projection and enable Scale to bounds in the redo panel (F6).
Add a material to the skydome and enable Shadeless under materials > Shading:


Add your texture with UV projection: